Describe all points in the complex plane that verify these inequalities: $| z + 3 i | \geq 3 | z − i |$ & $| z - 3/2 i | \geq 1$

Question

Describe all points in the complex plane that verify these inequalities:

$| z + 3 i | \geq 3 | z − i |$ & $| z - 3/2 i | \geq 1$

The answer is supposed to be the region between two disks of center $(0, 3/2)$ and of radius $1$ and $3/2$ respectively.

I tried squaring both sides of each inequalities but in vain. I have no idea how to get there.

Answer 1

Let $z=x+yi$

Hence $$x^2+(y+3)^2\ge 9x^2+9(y-1)^2$$ $$\Rightarrow 0\ge 8x^2+8y^2-24y=x^2+y^2-3y$$ Which is a disc with centre $(0,3/2)$ and radius $3/2$

And $$x^2+(y-3/2)^2\ge 1\Rightarrow x^2+y^2-3y+5/4\ge 0$$

Hence the answer follows